COLLEGE ALGEBRA FINAL MILESTONE
LATEST UPDATE 2024/2025
What is the value of the following expression?
16
-18
-2
0
RATIONALE
To solve this expression, evaluate the exponent for
each term. Start with the first term,
Any number taken to the power of zero equals 1.
So is equal to 1. Evaluate the next term, .
When the exponent is 1, the value of the term is the
same as its base, so 2 to the first power remains 2.
Next, evaluate the term which is the same as (-3)
(-3).
Negative 3 times negative 3 equals positive 9. The last
term, , indicates that -2 is multiplied by itself three
times.
equals -8 because when a negative number is
multiplied by itself an odd number of times, the
, answer remains negative. Finally, evaluate the
addition and subtraction from left to right with 1 minus
2.
1 minus 2 is -1. Next, add -1 and 9.
-1 plus 9 is 8. Finally add 8 and -8.
8 plus -8 is the same as 8 minus 8, or 0.
CONCEPT
Introduction to Exponents
2
To collect data on the signal strengths in a neighborhood, Tracy must
drive from house to house and take readings. She has a graduate
student, Dave, to assist her. Tracy figures it would take her eight hours
to complete the task working alone, and that it would take Dave 12
hours if he completed the task by himself.
How long will it take Tracy and Dave to complete the task together?
7.2 hours
6.0 hours
4.8 hours
3.6 hours
RATIONALE
We can use the relationship between work, rate, and
time to build equations to solve this problem. To
begin, we need to identify the relationship for Tracy
and Dave, separately.
Tracy's rate is 1 task in 8 hours. We do not know the
time it will take them to work on the project together,
so we will just denote the time as t. When we multiply
the rate and time together, we get Tracy's work
, at (notice how the units for hours cancel). We can
repeat this process for Dave.
Dave's rate is 1 task in 12 hours. We do not know the
time it will take them to work on the project together,
so we will just denote the time as t. When we multiply
the rate and time together, we get Dave's work at
. Now we can create a rational equation by adding
Tracy and Dave's work together.
The work is 1 task, so their combined work is equal to
1 task. In the rational equation, t is the amount of time
it takes Tracy and Dave to complete the task together.
With rational equations, we can add them together
once they have common denominators.
A simple way to find common denominators is to
multiply the denominators of the fractions together.
Make sure that you are doing this to all terms to
ensure they all end up with the same denominator.
Next, evaluate the multiplication.
Now we have three fractions with common
denominators. Once we have this, we can solve for t
using the expressions in the numerator.
To solve for t, first combine like terms.
12t plus 8t is equal to 20t. Finally divide both sides by
20 to solve for t.
It will take Tracy and Dave 4.8 hours to complete the
task together.
CONCEPT
Rational Equations Representing Work and Rate
3
What is the solution set for the following inequality?
4x – 3 + 6x > 5x + 7
x<2
x < 0.8
,
x > 0.8
x>2
RATIONALE
Before solving for x, make sure that each side of the
inequality is fully simplified. On the left side, we will
combine like terms 4x and 6x.
On the left side, 4x plus 6x equals 10x. Now we can
begin to solve for x by applying inverse operations to
both sides of the inequality. First, subtract 5x from
both sides of the inequality.
Subtracting 5x from both sides cancels the 5x term on
the right side of the inequality, leaving x terms on only
the left side. Next, add 3 to both sides of the
inequality.
Adding 3 to both sides leaves only the x term on the
left side. Finally, divide both sides of the inequality by
5 to isolate x.
The solution to this inequality is x > 2.
CONCEPT
Solve Linear Inequalities
4
The diameter of a proton is about meters. A hydrogen
atom has an overall length of 100,000 times (or times) the
diameter of a proton.
What is the length of the hydrogen atom, in meters, if it were written in
scientific notation?
meters
LATEST UPDATE 2024/2025
What is the value of the following expression?
16
-18
-2
0
RATIONALE
To solve this expression, evaluate the exponent for
each term. Start with the first term,
Any number taken to the power of zero equals 1.
So is equal to 1. Evaluate the next term, .
When the exponent is 1, the value of the term is the
same as its base, so 2 to the first power remains 2.
Next, evaluate the term which is the same as (-3)
(-3).
Negative 3 times negative 3 equals positive 9. The last
term, , indicates that -2 is multiplied by itself three
times.
equals -8 because when a negative number is
multiplied by itself an odd number of times, the
, answer remains negative. Finally, evaluate the
addition and subtraction from left to right with 1 minus
2.
1 minus 2 is -1. Next, add -1 and 9.
-1 plus 9 is 8. Finally add 8 and -8.
8 plus -8 is the same as 8 minus 8, or 0.
CONCEPT
Introduction to Exponents
2
To collect data on the signal strengths in a neighborhood, Tracy must
drive from house to house and take readings. She has a graduate
student, Dave, to assist her. Tracy figures it would take her eight hours
to complete the task working alone, and that it would take Dave 12
hours if he completed the task by himself.
How long will it take Tracy and Dave to complete the task together?
7.2 hours
6.0 hours
4.8 hours
3.6 hours
RATIONALE
We can use the relationship between work, rate, and
time to build equations to solve this problem. To
begin, we need to identify the relationship for Tracy
and Dave, separately.
Tracy's rate is 1 task in 8 hours. We do not know the
time it will take them to work on the project together,
so we will just denote the time as t. When we multiply
the rate and time together, we get Tracy's work
, at (notice how the units for hours cancel). We can
repeat this process for Dave.
Dave's rate is 1 task in 12 hours. We do not know the
time it will take them to work on the project together,
so we will just denote the time as t. When we multiply
the rate and time together, we get Dave's work at
. Now we can create a rational equation by adding
Tracy and Dave's work together.
The work is 1 task, so their combined work is equal to
1 task. In the rational equation, t is the amount of time
it takes Tracy and Dave to complete the task together.
With rational equations, we can add them together
once they have common denominators.
A simple way to find common denominators is to
multiply the denominators of the fractions together.
Make sure that you are doing this to all terms to
ensure they all end up with the same denominator.
Next, evaluate the multiplication.
Now we have three fractions with common
denominators. Once we have this, we can solve for t
using the expressions in the numerator.
To solve for t, first combine like terms.
12t plus 8t is equal to 20t. Finally divide both sides by
20 to solve for t.
It will take Tracy and Dave 4.8 hours to complete the
task together.
CONCEPT
Rational Equations Representing Work and Rate
3
What is the solution set for the following inequality?
4x – 3 + 6x > 5x + 7
x<2
x < 0.8
,
x > 0.8
x>2
RATIONALE
Before solving for x, make sure that each side of the
inequality is fully simplified. On the left side, we will
combine like terms 4x and 6x.
On the left side, 4x plus 6x equals 10x. Now we can
begin to solve for x by applying inverse operations to
both sides of the inequality. First, subtract 5x from
both sides of the inequality.
Subtracting 5x from both sides cancels the 5x term on
the right side of the inequality, leaving x terms on only
the left side. Next, add 3 to both sides of the
inequality.
Adding 3 to both sides leaves only the x term on the
left side. Finally, divide both sides of the inequality by
5 to isolate x.
The solution to this inequality is x > 2.
CONCEPT
Solve Linear Inequalities
4
The diameter of a proton is about meters. A hydrogen
atom has an overall length of 100,000 times (or times) the
diameter of a proton.
What is the length of the hydrogen atom, in meters, if it were written in
scientific notation?
meters
